D-Brane Models of Particle Physics Luis E. Iba´nez˜ · PDF fileD-Brane Models of...

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D-Brane Models

of Particle Physics

Luis E. IbanezUniversidad Autonoma de Madrid

SUSY-2002 Hamburg, June 2002

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INTRODUCTION

Dp-branes fundamental (p+1)-dimensional non-perturbative objects in

Type II string theory.

Dp

Gravity Gauge

Open strings () Gauge interactions

Closed strings () Gravity

Standard Model : Most obvious gauge theory to study:

LOOK FOR A D-BRANE DESCRIPTION OF THE SM

i.e., look for Dp-brane configurations in String Theory with a massless

spectrum resembling the SM.

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From a D-brane description of the SM one hopes:

� Unification of gravity and SM interactions

� Construct string realization of brane-world idea (perhaps)

But also to find new avenues to understand misteries of the SM like:

� Family triplication

� The unreasonable stability of the proton

� Fermion mass hierarchies

� etc.

A crucial requirement on the searched D-brane configuration is that it

should lead to

CHIRAL FERMIONS

D-branes on a smooth space leads to non-chiral theories (N = 4

SUSY)

Simplest ways to get chirality in explicit D-brane models:

� D-branes at singularities (e.g., orbifold sing.)

� Intersecting D-branes )

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� Talk based on:

L.I. , F. Marchesano and R. Rabadan, hep-th/0105155

D. Cremades, L.I, F. Marchesano , hep-th/0201205; hep-th/0203160;

hep-th/0205074

� Previous related work:

R. Blumenhagen, L. Gorlich, B. Kors, D. Lust, hep-th/0007024

G. Aldazabal, S. Franco, L.I., R. Rabadan, A. Uranga,

hep-th/0011073 ; hep-ph/0011132

R. Blumenhagen, B. Kors and D. Lust , hep-th/0012156

� also:

R. Blumenhagen, B. Kors, D. Lust, T. Ott, hep-th/0107138

M. Cvetic, G. Shiu, A. Uranga, hep-th0107166; hep-th/0107143

D. Bailin, G. Kraniotis, A. Love, hep-th/0108131

S. Forste, G. Honecker, R. Schreyer, hep-th/0008250

G. Honecker, hep-th/0112174; hep-th/0201037

C. Kokorelis, hep-th/0203187; hep-th/0205147

� brane intersections/fluxes :

C. Bachas, hep-th/9503030

M. Berkooz, M. Douglas, R. Leigh, hep-th/9606139

C. Angelantonj, I. Antoniadis, E. Dudas, A. Sagnotti, hep-th/0007090

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WHY INTERSECTING BRANES?

They have a number of properties present in the SM :

❶ Gauge group: Each stack of N branes carries a U(N) gauge

theory.

❷ Chirality: Two intersecting branes present chiral fermions at their

intersection, transforming in bifundamental (N; �M) or (N;M).

❸ Family replication: Branes at angles wrapping a compact manifold

may intersect several times.

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TRIPLICATION EXAMPLE

� Consider a pair of D4-branes

� 5-dimensional worldvolume = M4 � C1 , C1 wrapping a cycle on a

2-torus

e

e

T2

‘a’ and ‘b’ 4-branes intersect at three points

b b b

a

2

1

� wrapping numbers (n,m) = (1,0), (1,3)

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Varieties of Toroidal Intersecting Brane settings

Type II A, B String Theory

❶ D4-branes wrapping 1-cycles on T2 �R4=ZN

x

e

e

1

2

R / Z4

N

X4

T 2

❷ D5-branes wrapping 2-cycles on T4 �R2=ZNe

e

1

2

e 3

e 4

X

X2

R / ZN2

X

❸ D6-branes wrapping 3-cycles on T6

X

e

e

e

e

1

2 4

3 e

e

X

5

6

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Minimal Structure of SM D-brane settings

Configuration of 4 stacks of branes:

stack a Na = 3 SU(3)� U(1)a Baryonic brane

stack b Nb = 2 SU(2)� U(1)b Left brane

stack c Nc = 1 U(1)c Right brane

stack d Nd = 1 U(1)d Leptonic brane

R

L

LL

RE

LQ

U , D RR

W

gluon

U(2) U(1)

U(1)

U(3)

d- Leptonic

a- Baryonic

b- Left c- Right

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# Generations = # Colours ?

� Important constraint in ANY D-brane model with fermions in

bifundamentals (comes from RR-tadpole cancellation):

Number of N -plets = Number of �N -plets of U(N)

This is true even for U(2) or U(1).

� Impose Number of 2-plets = Number of�2-plets of U(2)

Left-handed SM fermions:

3 QL = 3 (3; 2) �! 9 2-plets

3 L = 3 (1; �2) �! 3 �2-plets

! Minimal SM has ‘U(2) anomalies’

6 extra fermion SU(2) doublets needed to cancel anomalies.

� Simple way to Cancel Anomalies :

2 (3; 2) + 1 (3; �2) �! 3 net 2-plets

3 (1; �2) �! 3 net �2-plets

�! U(2) anomalies cancel !!

� this works only because # COLORS = # GENERATIONS

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(N;M) and (N; �M) bifundamentals and orientifolds

We need (N;M) and (N; �M) bifundamentals to get the minimal

fermion spectrum of the SM

� In string theory they appear in orientifold models:

Orientifold = Type II / R

= worldsheet-parity ;R = some geometrical action (e.g.,

reflection)

� Under R :

Dp-brane ! Dp*-brane = ‘mirror’ = (R)Dp

Both brane and mirror need to be present in an orientifold

configuration

Dp Dp

Dp

_

a

b

a

Dp*b

(N , M ) (N , M )a b a b(1,-1) (1,1)

U(1) x U(1) chargesa b

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Quantum numbers of SM in intersecting brane models

R

L

LL

RE

LQ

U , D RR

W

gluon

U(2) U(1)

U(1)

U(3)

d- Leptonic

a- Baryonic

b- Left c- Right

Assuming all fermions come from bifundamentals and imposing

#N -plets =# �N -plets leads to the following model independent unique

structure (up to redefinitions):

Intersection Matter fields Qa Qb Qc Qd QY

(ab) QL (3; 2) 1 -1 0 0 1/6(ab*) qL 2(3; 2) 1 1 0 0 1/6(ac) UR 3(�3; 1) -1 0 1 0 -2/3(ac*) DR 3(�3; 1) -1 0 -1 0 1/3(bd*) L 3(1; 2) 0 -1 0 -1 -1/2(cd) NR 3(1; 1) 0 0 1 -1 0(cd*) ER 3(1; 1) 0 0 -1 -1 1

SU(3)� SU(2)� U(1)a � U(1)b � U(1)c � U(1)d

Where hypercharge is defined as:

QY =1

6Qa �

1

2Qc �

1

2Qd (1)

(Orthogonal linear combinations will be massive, see below)

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D6-BRANES WRAPPING AT ANGLES ON T6

Setup: type IIA D6-branes filling M4 and wrapping 3-cycles on T6.

We further assume having a factorized torus and factorizable 3-cycles:

T 6 = T 2 � T 2 � T 2

3-cycle = 1-cycle � 1-cycle� 1-cycle

X 2X 0

X 9

X 6 X 8X 4

X 7X 5

X 1 X 3

��

��

��

��

[�a] = (n1a;m1a)� (n2a;m

2a)� (n3a;m

3a)

Intersection number: Iab = I1ab � I2ab � I3abIiab =

�niam

ib �mi

anib

�

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ORIENTIFOLD

Consider the theory

Type IIA on T 6

RT�dual ! Type I on ~T 6 (2)

: Worldsheet parity.

R = R(5)R(7)R(9)

Consequences:

❶ only some tori lattices are allowed by R action: square (b(i) = 0)

or tilted (b(i) = 12

).

❷ Mirror branes should be added

For each D6-brane a in our model, we must add its mirror image a�

under R.

Deffine effective wrapping numbers as

(nia;mia)e� � (nia;m

ia) + b(i)(0; nia),

Then R action reduces to

D6� brane a 7! D6� brane a�

(nia;mi

a) (nia;�mi

a) (3)

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❸ Fermion spectrum

� D6aD6a and D6a �D6a� sectors : U(N)N = 4 SYM.

a

a

� D6aD6b and D6aD6b� sector: bifundamental fermion

representations Iab(Na; Nb) + Iab�(Na; Nb)

b

a

b*

a

(asuming each brane does not intersect with its own mirror, which

would lead to additional exotic symmetric/antisymmetric fermion

fields )

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RR Tadpole Cancellation Conditions

� Consistent compactifications must satisfy RR Tadpole conditions,

which imply a vanishing total Charge under certain

Ramond-Ramond antisymmetric fields. This automatically ensures

non-abelian SU(Na)3 anomaly cancellation:

PcNcIac = 0.

Notice that, as advertised, in these systems SU(N)3 anomaly

cancellation reduces to

# fundamentals = # antifundamentals

� If D6-branes have wrapping numbers (n1a;m1a)(n

2a;m

2a)(n

3a;m

3a)

the conditions read:

X

a

Na n1an

2an

3a = 16 (4)

X

a

Na n1am

2am

3a = 0

X

a

Na m1an

2am

3a = 0

X

a

Na m1am

2an

3a = 0

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SM INTERSECTION NUMBERS

Iab = 1 ; Iab� = 2

Iac = �3 ; Iac� = �3

Ibd = 0 ; Ibd� = �3

Icd = +3 ; Icd� = �3

a = U(3)baryon ; b = U(2)left

c = U(1)right ; d = U(1)lepton

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Getting JUST the SM at intersecting D6-branes

Look for choices of wrapping numbers (nia;mia) yielding the fermion

spectrum of the SM that we displayed before:

The general solution is:

Ni (n1i;m1

i) (n2

i;m2

i) (n3

i;m3

i)

Na = 3 (1=�1; 0) (n2a; ��2) (1=�;�~�=2)

Nb = 2 (n1b; �~��1) (1=�2; 0) (1;�3�~�=2)

Nc = 1 (n1c ; 3��1) (1=�2; 0) (0; 1)

Nd = 1 (1=�1; 0) (n2d; ��2=�) (1; 3�~�=2)

Table 1: D6-brane wrapping numbers giving rise to a SM spectrum. The general

solutions are parametrized by two phases �; ~� = �1, the NS background on

the first two tori �i = 1 � bi = 1; 1=2, four integers n2a; n1b ; n

1c ; n

2d and a

parameter � = 1; 1=3.

Tadpole conditions satisfied if:

3n2a��1

+2n1b�2

+n2d�1

= 16 : (5)

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U(1) symmetries


(ab) QL (3; 2) 1 -1 0 0 1/6(ab*) qL 2(3; 2) 1 1 0 0 1/6(ac) UR 3(�3; 1) -1 0 1 0 -2/3(ac*) DR 3(�3; 1) -1 0 -1 0 1/3(bd*) L 3(1; 2) 0 -1 0 -1 -1/2(cd) NR 3(1; 1) 0 0 1 -1 0(cd*) ER 3(1; 1) 0 0 -1 -1 1

SU(3)� SU(2)� U(1)a � U(1)b � U(1)c � U(1)d

❶ Qa = 3 B ; Qd = L ; Qc = 2IR

These known global symmetries of the SM are in fact gauge

symmetries !!

❷ Two are anomaly-free :

Qa

3� Qd = B � L (6)

Qc = 2IR

with

Y = 12(B � L) � IR

❸ Two have triangle anomalies:

3Qa + Qd ; Qb

Anomalies are cancelled by a Generalized Green-Schwarz

mechanism

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Green-Schwarz mechanism

i

Bµνi η i

duals

U(1)a

U(N )b

U(N )

U(N )b

U(N )b

b

B

B^Fai

F^Fηi b b

c ai

dib

ic a d

ibΣ

ι + Aa

b = 0

� If a U(1) is anomalous it is necessary massive, due to the B ^ Fa

couplings:

��B��(@�A�) = (@��)A

� = Higgs � like coupling

� but not the other way round.

� In above orientifold models, there are 4 D = 4 RR fields involved,

thus at most 4 U(1)’s can gain mass.

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� Even if the abelian gauge symmetry is lost, the U(1)’s remain as

perturbative global symmetries.

� In our specific solutions, there are two model-independent

anomalous U(1)’s, and only one U(1) remains massless:

Q0 = n1c(Qa � 3Qd) �3~��2

2�1(n2a + 3�n2d)Qc (7)

It coincides with standard hypercharge if:

n1c =~��2

2�1(n2a + 3�n2d): (8)

) Just the SM group and 3 generations

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Scalars at D6-brane intersections

θi

i=1,2,3 toriab

a

b

� There are three lightest scalars (“squarks/sleptons”) at eachintersection with masses in string units:

M21 = 1

2(�j#1j+ j#2j+ j#3j)

M22 = 1

2(j#1j � j#2j+ j#3j)

M23 = 1

2(j#1j+ j#2j � j#3j)

(9)

� For wide ranges of parameters scalars are non-tachyonic

� For particular choices of radi and wrappings nia;mia there is a

massless scalar, signaling the presence of N = 1 SUSY at THAT

intersection

� A fully N = 1 SUSY toroidal brane configuration in which all

intersections respect the same supersymmetry is not possible (due

to RR-tadpole cancellation).

� But one can obtain configurations in which all intersections respect a

DIFFERENT N = 1 SUSY) Q- SUSY models.

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A SM with different SUSY at each intersection

Consider the particular subset of models with wrapping numbers

Ni (n1i;m1

i) (n2

i;m2

i) (n3

i;m3

i)

Na = 3 (1; 0) (n2a; �2) (3;�1=2)

Nb = 2 (n1b; 1) (1=�2; 0) (1;�1=2)

Nc = 1 (0; 1) (1=�2; 0) (0; 1)

Nd = 1 (1; 0) (n2a; 3�2) (1; 1=2)

and verifying

U1 =n1b2U3 ; U2 =

n2a6�2

U3 where U i = Ri2=R

i1 ; i = 1; 2; 3

� One can check quarks and leptons have massless SUSY partners

with respect to 4 different SUSY’s:

N=1 N=1

N=1 N=1

baryonic

leptonic

left right

R RU ,DQL

L E , NR R

N=1N=1N=1N=1

N=4

� Interesting class of theories in which quantum corrections to scalars

appear only at two loops

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N=4

N=1

N=1’ N=0

N=1

a) b)N=0

N=4 N=4

� May help in stabalizing a modest hierarchy in between the weak

scale and a low string scale Ms / 10� 100 Tev

Getting the chiral spectrum of the MSSM

� It is also possible to get a D6-brane configuration with the chiral

spectrum of the MSSM and quarks, leptons and Higgs multiplets

respecting the same N = 1 SUSY

Q U , D

L E , N

L R R

R R

H , HU D

Baryonic

Leptonic

Left Right

U(3)

U(1)

U(2) U(1)L

R

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brane type Ni (n1i;m1

i) (n2

i;m2

i) (n3

i;m3

i)

a2 Na = 3 (1; 0) (3; 1) (3;�1=2)b2 Nb = 2 (1; 1) (1; 0) (1;�1=2)c2 Nc = 1 (0; 1) (0;�1) (2; 0)a2 ’ Nd = 1 (1; 0) (3; 1) (3;�1=2)

Table 2: Wrapping numbers of a three generation SUSY-SM withN = 1 SUSY

locally.


ab QL (3; 2) 1 -1 0 0 1/6ab� qL 2(3; 2) 1 1 0 0 1/6ac UR 3(�3; 1) -1 0 1 0 -2/3ac� DR 3(�3; 1) -1 0 -1 0 1/3bd L (1; 2) 0 -1 0 1 -1/2bd� l 2(1; 2) 0 1 0 1 -1/2cd NR 3(1; 1) 0 0 1 -1 0cd� ER 3(1; 1) 0 0 -1 -1 1bc H (1; 2) 0 -1 1 0 -1/2bc� �H (1; 2) 0 -1 -1 0 1/2

Table 3: Chiral spectrum of the SUSY’s SM obtained from the above wrapping

numbers with U1 = U2 = U3=2.

� The model is not fully N = 1 SUSY because to cancel

RR-tadpoles additional massive N = 0 sectors (with no intersection

with SM ones) have to be added. Also N = 4 gauge sector.

� Due to these additional N = 0 sectors the model looks somewhat

like a gauge mediated SUSY-breaking model.

� There is an additional U(1)B�L.

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Higgs mechanism and brane recombination

❶ Brane separation = Adjoint Higgsing Does not lower the rank.

U(N) U(N-1)xU(1)

N Dp-branes

❷ Brane recombination lowers the rank

Tachyon H H=0

U(1)xU(1) U(1)b f

b f

f

c

c

c+b = f

In SM the rank is lowered ! brane recombination of the branes b and

c(c�) at which intersection the Higgs scalars lie.

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Hierarchical Yukawa couplings

� Yukawa couplings come from triangular worldsheets

��

��

��

��

��

��

Q q

H <H>=0

Q qL L R R

= exp (- Area) m = <H> exp(-Area)Y

b c

a

f

a

f

� Different distance to Higgs field gives rise to hierarchical Yukawas (

Aldazabal, Franco, L.I., Rabadan, Uranga , hep-ph/0011132 ).

u u

t t

c c

H

L

L

R

R

L R m >>m >> mt c u

baryon

baryon

baryon

left right

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Getting MP lanck >> Mstring

Non-SUSY models: To avoid hierarchy problem one should have

Ms / 1 TeV (Arkani-Hamed,Dimopoulos,Dvali)

❶ D6-branes The A-D-D approach for MPlanck >> Mstring by

making transverse volume large not possible: no tori direction

transverse to all D6-branes simultaneously. CY? Warping?

❷ D5-branes Analogous intersecting models can be built yielding just

the SM fermion spectrum. These are Type IIB compactified on e.g.,

T 2 � T 2 � (T 2=ZN ) with D5-branes wrapping 2-cycles on

T 2 � T 2. In this case

MPlanck = M2string

1

�

pV2

a

b

d

c

q L

q RL

E R

1Z

Z2

x2 2T T

B2

Z3

D5-brane

Singularity

for large

PlanckM

String>> M

2V = Vol(B )2

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Gauge coupling constants

SU(3)� SU(2)� U(1)a � U(1)b � U(1)c � U(1)d

� Gauge couplings are not unified :

1

g2i

=M3

s

(2�)4�Vol(�i) ; i = a; b; c; d (10)

Vol(�i) being the volume each D6-brane is wrapping.

� Thus, e.g., SU(3) interactions are stronger than SU(2) because

‘baryonic’ branes wrap less volume than ‘left’ branes

� There are 6 groups but only 4 couplings) There are some

relationships :

g2a =g2QCD6

; g2b =g2L4

;1

g2Y=

1

36g2a+

1

4g2c+

1

4g2d: (11)

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TeV-Scale Z’ Bosons from D-branes

� There

is quite a generic structure of extraU(1)’s. They are NOT ofE6 type:

(B � L) ; IR ; (3B + L) ; Qb

� If Ms / 1� 10 TeV , could perhaps be tested at present/future

accelerators

� Extra Z ’s get masses by combining with RR string fields B��i

2

= Σi string

2

U(1) U(1)a bB

µνi

c B ^ F c B ^ Fa

a b

bi i i i

Mab a b

a bg g c c Mi i

� Four Eigenvalues = (0;M2;M3;M4). In D5, D6-brane models one

finds typically at least one of them M3 <13Mstring

� Three massive Z ’s mix with SM Z0 . One can put constraints on

Mi from the �-parameter. (D. Ghilencea, L.I., N. Irges, F.Quevedo,

hep-ph/0205083.)

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Baryon and lepton number violation

� Baryon number is a gauge symmetry. So the proton is automatically

stable. (Baryogenesis should take place non-perturbatively).

� Lepton number is also a gauge symmetry. But may be

spontaneously broken (e.g. ~�R vevs.). In the first case only Dirac

masses. In the second case Majorana masses of order Mstring

possible.

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CONCLUSIONS

❶ Intersecting D-brane constructions provide for an understanding

from the string theory point of view of questions like

� chirality

� family triplication

� proton stability

❷ Certain phenomena of the SM have a geometrical interpretation:

� Hierarchical Yukawas come from the different areas of triangles

connecting the Higgs with left and right fermions.

� Different sizes of gauge couplings come from different volumes

wrapped by the different branes.

� The SM Higgs mechanism has a geometrical interpretation as

brane recombination

❸ We have constructed classes of Intersecting D6 and D5-brane

models wrapping toroidal compactifications in (orientifolded) Type II

strings.

� The minimal intersecting D-brane SM constructions are obtained

from 4 stacks of branes: Baryonic, Leptonic, Left and Right.

� Some classes of D6 and D5 brane models provide the first

explicit string constructions with just the fermions of the SM and

gauge group SU(3)� SU(2)� U(1)Y .

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� The SM intersection numbers are topological in character and

may appear in general D-brane configurations wrapping e.g.,

CY-compactifications.

� It is also possible to find D6-brane configurations with the chiral

spectrum of the MSSM, although there is also a massive N = 0

sector in these toroidal examples.

� Note that these ideas are not necesarily tied up to a low string

scale scenario.

some homework...

� New configurations of Dp-branes wrapping general cycles of e.g.,

CY3 and same intersection structure. (see recent R. Blumenhagen,

V. Braun, B. Kors, D. Lust, hep-th/0206038.)

� Stability of configurations. The models discussed are non-SUSY and

generically have NS-tadpoles. Look for stabilized vacua (e.g. with

antisymmetric tensor fluxes). (Constructing N = 1 models is nice

but once SUSY is broken stability issues will have to be faced

anyhow).

� Phenomenology of D6 and D5 models :

– Try to reproduce quark/lepton masses and mixings from

hierarchical Yukawas.

– Study of signatures of three ‘canonical’ extra Z ’s.

� ............

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